Computational Modeling of Oscillatory Motion with Finite Difference Time Domain Method. Yue Wu

Transcription

1 Computational Modeling of Oscillatory Motion with Finite Difference Time Domain Method by Yue Wu A Report Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF ENGINEERING in the Electrical and Computer Engineering Yue Wu, 217 University of Victoria All rights reserved. This report may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author.

2 i Abstracts Finite Difference Time Domain (FDTD) method is a numerical analysis technique used in computational electromagnetics. Since it was proposed in 1966, FDTD method has become the fastest growing and the most popular method compared to other numerical solutions. Nowadays, using finite difference time domain method to model stationary object has been well established, there are several commercial and open-source FDTD solutions available on the market, which are ecellent at modeling immobile devices. But little research has been performed to study the modeling of dynamic movements such as vibration and oscillation, which can be especially useful for studying deformations caused by forces like radiation pressure to better understand interactions between matter and electromagnetic waves. In this project, a two-dimensional FDTD model with an oscillating cylindrical rod was proposed and implemented. Using this model, the Raman scattering effect caused by an oscillation device was successfully observed. And a further investigation about the enhancement of Raman scattering when the incident frequency is near a whisper gallery mode resonance was performed. A minimal resonance shift caused by the oscillatory motion was also observed.

3 ii Table of Contents 1. Introduction Background Literature Finite Difference Time Domain Method Mawell s Equations Yee Algorithm Yee Grid Update Equation for EE and HH Absorbing Boundary Conditions and Perfectly Matched Layer Update Equation with PML Other Conditions for FDTD Simulations Source Ecitations Total-field/Scattered-field Formulation Correction Terms for TF/SF FDTD with an Oscillating Object Raman Effect Implementing Oscillating Device Update Equation with Time-varying Property Transitional-layer Results Raman Sidebands Whispering-gallery Mode Resonance Raman Shift near WGM Resonance Conclusions and Future Work References Appendi Pre-loop Computation FDTD Loop Oscillation Function... 51

4 1 1. Introduction Since Mawell established the governing equations for electromagnetic fields in 1873, electromagnetic theory and related researches have been over a hundred years. At present, the theory of electromagnetism has been widely applied in various fields like radio propagation, optical communication, antenna design, optical imaging, spectroscopy and so on. Propagation of electromagnetic fields in real environments can be very complicated, such as scattering from microscopic structures, radiation of comple antennas, propagation in waveguides, etc. With so many different applications, generalized computational models become increasingly important to give more insights on eperiments and to verify eperimental data with simulations. Many meaningful numerical solutions to Mawell's equations have been proposed such as method of moments (MOM), finite element method (FEM), and finite difference time domain method (FDTD) etc., of which FDTD is becoming the fastest growing and the most popular method in use today. Nowadays signal scattering from a stationary object can be well simulated using FDTD method. As interests grow in understanding how matter interacts with electromagnetic waves, the analysis of the electromagnetic field around dynamic bodies also has received great interests. But few studies have used FDTD methods to study the modeling of dynamic motion such as vibration and oscillation, which can be especially useful for microscopic deformations caused by forces like radiation pressure. The purpose of this project is to develop a model to simulate two-dimensional harmonic oscillation with finite difference time domain method, a custom FDTD program was implemented since no commercially available FDTD solvers provide tools to integrate the necessary modifications required for this task. This report is organized as follows: First, a review of literature that incorporates dynamic systems with numerical models is presented

5 2 in Section 2. In Section 3, a compressive review of FDTD method and its formulation is discussed, the corresponding code will be available in the appendi. Net in Section 4, a brief introduction of Raman scattering effect which is the main phenomenon could be observed with an oscillating object was presented. The methodology to incorporate an oscillating cylinder in our FDTD algorithm is also outlined. Then in Section 5, some simulation results based on the developed model are presented. And finally, Section 6 will be devoted to concluding remarks and a discussion of future works.

6 3 2. Background Literature Although there is no conclusive numerical model yet, radiation pressure is widely considered to be the connection between electromagnetic and mechanical systems. A simulative eperiment done by Ma Waddell and Kenneth Chau [1] tried to incorporate different sets of electrodynamic postulates to model radiation pressure. In their research, they implemented electromagnetic and mechanical systems separately with FDTD to simulate the electromagnetic fields, then compute power, energy, stress, and momentum from the results based on different postulates and use them to simulate the kinetic moment with Newtonian dynamics. However, in their simulations, they assume that electromagnetic wave causes negligible displacement of the material, so the mechanical system was reduced to a center of mass analysis. This would not be a valid assumption for small particles and molecules since they will be oscillating under electromagnetic radiation. As demonstrated by Mihiretie, in their eperiments [2,3] they observe ellipsoidal dielectric particles with higher refractive indices compared to the surrounding fluid, trapped around the laser beam and constantly vibrating, rotating, and oscillating. But despite observing the rotation and oscillation, their simulative analysis also only considers the spatial vibration and ignore the other two. The reason they ignore oscillations is due in part to the fact that there are currently no established methods to model vibrating and oscillating object with electromagnetic fields. A few pieces of literature eist with oscillation been implemented in simulation techniques other than FDTD method, but these implementations have some major compromises. As a time-domain method, FDTD is inherently a better and more intuitive way to handle dynamic systems. To incorporate comple motion, such as oscillation, we first need to look

7 4 back to the evolution of modeling dynamic objects with finite difference time domain method. It starts with modeling time-varying medium using FDTD method, the term time-varying means the property of the medium can be altered as a function of time independent of the electromagnetic field values. The theoretical studies of time-varying medium often found that the solutions of Mawell s equations are etremely difficult to obtain in analytical forms, and the solutions obtained can only be applied to idealized problems. Taylor, Lam and Shumpert [4] were the first to prove this can be solved by FDTD method. In particular, they used FDTD algorithm to eam the electromagnetic pulse scattered from a cylindrical rod inside a cylindrical waveguide, the conductivity of the cylindrical rod was assumed to be linearly increasing as a function of time, but the permittivity was kept constant. Harfoush [5] applied FDTD solutions to analyzes electromagnetic wave penetrated and scattered from a material with its conductivity time-varying sinusoidaly. Comparisons of the FDTD solutions to their analytical results were ecellent. Liu [6] have presented literature dedicated to investigating the stability of FDTD method for modeling timevarying permittivity by comparing the simulation results with previous results obtained using theoretical approaches in other works. One simulation was done for plane wave interaction with a dielectric slab that has sinusoidal time-varying permittivity, and the other was for a microstrip patch antenna on a substrate with a sinusoidal time-varying permittivity. The stability of the FDTD solution in their case proved to be very promising. The author also mentioned that by changing the permittivity sufficiently slowly with respect to the frequency of the incoming wave, the permittivity variation can be considered time-invariant, and the result becomes more accurate.

8 5 After proving that the problem of time-varying properties can be solved using finite difference time domain method, several numerical implantations have been proposed to model uniform translational motions with the FDTD method. The first time FDTD was ever used in modeling electromagnetic wave scattered by moving object was done by the research group of Allen Taflove. In their paper [7], Harfoush introduced a numerical approach to deal with the electromagnetic wave scattering properties for moving or vibrating objects. The movement of the object was realized by relativistic boundary condition. However, the relativistic boundary condition was only adopted for perfect conducting mirrors, that is objects only receding the incident wave without transmission or refraction. Also, the vibration in his paper was merely two translational motion in opposite directions since the mirror was not deformed. The author also mentioned that by considering a sufficiently small velocity, the ratio between the object s velocity and speed of light vv/cc could then be ignored. Thus, the formulation of the relativistic boundary condition became a total reflection function with the field reflected by the boundaries being doubled. Which indicate that the boundary condition is not as important in low velocities as in high-speed scenarios. Recently Inman et al. [8] reported a method to realize a constant speed movement by using dielectric approimation and intermedia-step field movement. On the forward-boundary of the moving object, dielectric coefficients are modified at each step until they possess the properties of an inside cell. Likewise, on the back-boundary cell, they are modified until they become free space properties. After the coefficients are modified, the fields inside and outside the boundary are split in two as total-field and object-field. The total-field outside the boundary is a summation of the background field without the object s present and the

9 6 object-field that was assumed as a portion of the field that eists only within the moving object. The ratio of the object-field versus the total-fields was assumed to be governed by the permittivity of the medium. Then the object-field was also considered to be spatially shifted, therefore, a correction term was calculated with a Lagrange approimation in the direction of the movement, however, this correction term is only feasible for objects moving in one direction. And won t be necessary at all if the movement is sufficiently slow compared to the speed of the light, renders the shift insignificant. Inman s method relied heavily on precomputed lookup tables since the entire coefficient approimation and Lagrange approimation was precomputed which can be easily done in one-dimensional cases and two-dimensional cases with only one directional movement but become dramatically more difficult to generalize for off-ais translational motion or vibrations in two dimensions. A more recent technique proposed by Hiroshi Iwamatsu [9] combines Lorentz transformations with FDTD method was dedicated to analyzing interactions with high velocities close to the speed of light. The moving devices were modeled on a secondary sub-grid and can be considered static in perspective of the sub-grid, but the sub-grid was moving relative to the main grid. Since the object is stationary in the perspective of its own grid, regular FDTD simulations can be performed separately in each grid, then the fields need to be interpolated between two grids with Lorentz transformations. Considering two set of grids need to be simulated at the same time, this approach is incredibly computationally demanding compared to normal FDTD algorithms. In general, these are the three main methods for molding moving bodies with the FDTD method currently proposed. But the interests of these methods were mainly focused on

10 7 high-speed moving bodies, and most of the techniques could not be transferred to model vibration and oscillation. Currently, there is no meaningful literature tried to incorporate vibration and oscillation with FDTD methods alone. Although few attempts have been made to simulate them with other numerical methods. Murray [1] introduce an analytical theory to calculate the polarizability of oscillating nanoparticles. The vibrational motion was described by spherical harmonic functions. Since his entire method was built on a Spherical coordinate system with no time domain element, it would be difficult to adopt it in FDTD. The author also states that FDTD cannot handle very small changes in the object since it must use a coarse grid of spatial points which touches on one major drawback of the technique, that is small shapes are difficult to be accurately represented by rectangular grids. This can be solved by either a higher resolution or alternative grid systems like heagonal grids or curvilinear grids, etc. However, either increasing the resolution or using alternative grid systems will also result in requiring considerably more computational resources. It is worth noting that Murray also points out the permittivity of the ellipsoid in his method needs to be smoothed. But different to Inman s approach by smooth the permittivity only on the boundary over time, he smoothed the permittivity over a short distance near the surface, which alters the effective radius of the nanoparticle. A drawback that can be easily avoided with time domain methods. To understand how waves scattered from metal nanoparticles respond to vibrations, Ahmed used FDTD calculations only for the optical response in his paper [11]. The structural changes were modeled by solving the Navier equation with the continuity of stress and displacement at the boundary using the finite element method. In their case, the vibration

11 8 of a two-dimensional nanowire was molded by a cylindrical nanorod with fied width. The deformation was done by changing the length of the rod, which somehow represents an oscillation ellipse. But since the width is fied, the vibration is essentially two translational motion in opposite directions, similar to what Harfoush did with his mirror vibrations. The boundary condition was not addressed at all in this paper, however, the vibrational speed described in the paper was considerably slow compared to the speed of light. Thus, giving us more reason to believe that the boundary condition may not be as important for simulations with ultra-slow velocities. Methods to incorporate vibrations were also implemented in cavity optomechanical crystals analysis by research groups of Vahala and Painter in recent years. As showcased by Eichenfield [12], they use a Finite element method to compute both electromagnetic and mechanical systems, the mechanical displacement profile is defined to describe the perpendicular displacements of volume elements. Optical and mechanical mode volumes are defined by electric field and mechanical displacement profile to gauge the strength of light-matter interactions. Then vibrations were introduced via a variable effective length coefficient calculated by optical, mechanical mode volumes and electric field values, based on a first-order perturbation theory of Mawell s equations that was described in [13]. This approach is an elegant way to incorporate deformation, but also has its own limitations. For eample, the perturbation theory only possesses the first-order accuracy and usually does not handle nonlinear effects well.

12 9 3. Finite Difference Time Domain Method This section presents a review and formulation of the finite difference time domain methods as well as formulations of different techniques relate to the FDTD that is used in this project, including perfectly matched layer boundary conditions, Courant-Friedrichs- Lewy stability conditions, Gaussian pulses, plane wave ecitations and totalfield/scattered-field techniques, etc Mawell s Equations In general, electric and magnetic fields can be represented by vector quantities that have both magnitude and direction. The behavior of electric and magnetic fields generated by electric charges or a flow of electric current are governed by physical laws known as Mawell s equations which can be epressed in a set of partial differential equations as: Gauss s law: D = ρ (3.1) Faraday s law: Ampere s law: B = (3.2) B E = t D H = + J t (3.3) (3.4) Where DD is electric flu density, ρρ is electric charge density, BB is magnetic flu density, EE is electric field intensity, HH is magnetic field intensity, and JJ is electric current density. The response between DD and EE, as well as BB and HH are specified through constitutive relations, for linear materials, they are: D= E = E [ ε] ε [ ε ] r B= H = H [ µ ] µ [ µ ] r (3.5) (3.6)

13 1 Where [εε] is the permittivity of the material that equals to free-space permittivity constant εε multiply by the relative permittivity tensor [εε rr ]. Similarly, the permeability of the material [μμ] is the product of free-space permeability μμ and relative permeability [μμ rr ]. Electric current density JJ can be estimated with conductivity [σσ] through Ohm s law: J = [ σ ] E (3.7) In linear, non-dispersive materials, Mawell s equations can be represented with only electric and magnetic fields as: H E = µ [ µ r ] t E H = ε [ εr ] + [ σ] E t (3.8) (3.9) 3.2. Yee s Algorithm In 1966 Kane S. Yee [14] proposed the Finite Difference Time Domain algorithm which is a numerical solution of Mawell s equations. By applying a set of central-difference approimations, the spatial and temporal derivatives appearing in Mawell s equations can be estimated with second-order accuracy. Consider a function ff, its Taylor-series epansions at points ± δδ are: δ δ 1 δ ( 2 ( ) ( ) ) 1 δ 3 ( ) ( f ) + = f + f + f + f ( ) ! 2 3! 2. (3.1) 2 3 δ δ 1 δ ( 2 ( ) ( ) ) 1 δ 3 ( ) ( f ) = f f + f f ( ) ! 2 3! 2 (3.11) Subtracting (3.11) from (3.1) then divided by δδ yields:

14 11 δ δ f + f δ = ( 3 f ( ) ) + f ( ) + (3.12) δ 3! 2 Therefore, the left-hand side of the equation equals to the derivative of ff at plus OO(δδ 2 ). Provided δδ is sufficiently small, a reasonable second-order approimation for an arbitrary point could be given by: δ δ f + f f ( ) 2 2 δ = (3.13) Note that the approimation is not sampled at, but at its neighboring positions ± δδ 2 instead. To accommodate this, Yee also proposed a simple yet elegant rectangular grid scheme that staggers the electric and magnetic fields both spatially and temporally. This scheme now known as Yee grid has proven to be very robust and become the core of modern FDTD algorithms Yee Grid Consider a three-dimensional rectangular grid, each grid cell has the length of, yy and zz along each ais, and a time difference of tt to its adjacent cells. A notation ff ii,jj,kk (nn) can epress any cell in the grid with: ( i, jk, ) ( i, j yk, z) = (3.14) And: n= ( n t) (3.15) Within this grid system, vector components of the electric field are projected parallel to the edge of the cells and are sampled at the center of each edge. Vector components of the magnetic field are projected normal to the faces of the cell and are sampled at the center of each face as illustrated in Figure.1a. Note that each EE component and its surrounding HH

15 12 components have a half spatial difference as well as a half temporal difference, vice versa, each HH component with its surrounding EE components also has a half spatial and temporal differences demonstrated in Figure.1b-Figure.1c. Figure 1. a. standard 3-D Yee cell with perspective of the spatial locations of EE and HH components. b. 2-D Yee cells in transverse-magnetic (TM) mode with perspective of positions between Ez component and HH components. c. 2-D Yee cells in transverse-electric (TE) mode with perspective of positions between Hz component and EE components. 3.4 Update Equation for EE and HH This grid system works perfectly with our central-difference approimations and provides essential pieces for formulating the curl operations in Mawell's equations which can be epressed as: E E z y E E Ez y E E = ˆ + ˆ + ˆ y z z y z y H H z y H H Hz y H H = ˆ + ˆ + ˆ y z z y z y (3.16) (3.17) For purpose of this project, the formulations will be reduced to two-dimensional TM mode via assuming partial derivatives of fields with respect to zz detention equal to zero. And only preserve the zz component of the electric field, and yy components of the magnetic field. The simulative results of the TM and TE mode have proven to be identical. Also CCCC

16 13 and CCCC yy is denoted as the curl of EE in, yy directions, CCCC zz as the curl of HH in zz direction. Using spatial central-difference approimations for each derivative in the curl operations, for point (ii, jj) at time tt the vector epansion of the curl operations becomes: CE CE CH i, j i, j y i, j z ( t) ( t) ( t) i j ( ) ( ) i, j+ 1, E E z y Ez t Ez t = = y z y ( ) ( ) i+ 1, j i, j E E z z t Ez t E = = z ( ) ( ) ( ) ( ) i, j i 1, j i, j i, j 1 Hy H H y t Hy t H t H t = = y y (3.18) (3.19) (3.2) Note that CCCC ii,jj eists at the same spatial position as HH ii,jj which is located at EE ii,jj+1/2 zz, same goes for CCCC ii,jj yy eists at EE ii+1/2,jj zz and CCCC ii,jj zz locate at the same position as HH ii,jj 1/2 yy and HH ii 1/2,jj, such being the case, in Eq. (3.12) correspond to jj + 1/2, ii + 1/2, ii 1/2, jj 1/2 in Eq. ( ) respectively. Before proceeding to the right side of Mawell s equations in Eq. ( ), the boundary conditions should be discussed first, since after implementing loss into the boundary, some artificial property will be added that changes the formulation. 3.5 Absorbing Boundary Conditions and Perfectly Matched Layer (PML) When running FDTD simulations, a boundary must be artificially added that will terminate the grid via effectively absorbing any electromagnetic field that travels beyond the problem domain. Such a boundary is referred to as the absorbing boundary condition (ABC). Traditional ABC methods are etremely computationally demanding and still reflect a noticeable amount of energy back to the computational domain [15]. The 9s saw the emergence of absorbing media ABCs, including some powerful methods as Berenger s

17 14 original perfectly matched layer (PML) [16] and later a more efficient and more popular uniaial PML (UPML) [17] [18]. The property of the PML region is selected such that waves incident upon the PML region do not reflect at the interface, meanwhile eponentially decaying inside the PML region. The PML region is considered to be composed of an anisotropic material that has comple diagonal permittivity and permeability tensors in the form of: σ µ + jω σ y [ µ ] µ y + = [ µ r] = jω µ σ z µ z + jω σ ε + jω σ y εy + = [ εr] = jω ε σ z ε z + jω [ ε ] (3.21) (3.22) Where εε,yy,zz, μμ,yy,zz, σσ,yy,zz are properties eclusive to the PML region. The loss is incorporated by a fictitious conductivity mimicking loss caused by real conductivity. In order to incorporate loss without introducing additional reflection from the change of impedance, the impedance of the PML required to be matched to free space: η µ µ r = = (3.23) ε ε Which implies that [εε rr ] = [μμ rr ], thus our diagonal tensors in Eq. ( ) can be epressed in the same form as:

18 15 [ S] a [ µ ] [ ε] = = = b µ ε c (3.24) In their paper Sacks et al. [18] showcase that the refraction and reflection can be manipulated through the choice of aa, bb and cc. Consider the problem of a wave entering the PML in zz direction represented in Figure.2, according to [19] the Snell s law for diagonally anisotropic media can be epressed in the form: sinθ = sinθ (3.25) i r sinθ = bc sinθ (3.26) i t Reflection coefficients for the TM and TE modes rr TTTT and rr TTTT in Fresnel equations can be written as: r TM = acosθ acosθ + i i bcosθ t bcosθ t (3.27) r TE = bcosθ acosθ + t i acosθ i bcosθ t (3.28) Figure 2. Reflection and refraction represented when wave inter PML interface on a z plane, θθ ii is the angle of incident wave, θθ rr is the angle of reflection, θθ tt is the angle of refraction.

19 16 By imposing the condition bbbb = 1, the refraction will be removed by forcing θθ ii = θθ tt, cos θθ ii and cos θθ tt can also be canceled out, which indicates the reflection will no longer be a function of incident and transmission angles. By further imposing the condition aa = bb, both reflection coefficients must be zero. This property is completely independent of the incidence angle, polarization or incident frequency. Therefore, for any wave traveling in zz direction, our PML tensor should have aa = bb = cc 1, which makes the PML property uniaial anisotropic. And the diagonal tensor [SS] in zz direction can be written as: [ S ] z sz = sz 1 s z (3.29) Similar to wave entering in the zz direction, for waves traveling in and yy directions should have aa 1 = bb = cc and aa = bb 1 = cc respectively. Combining three, yy, zz directional tensors into a single tensor quantity results in: 1 s ss y z 1 S = S Sy Sz = ss y sz 1 sss y z [ ] [ ] [ ] (3.3) Where ss,yy,zz = 1 + σσ,yy,zz /jjjjεε, which is the most popular choices for implementing loss in many PML implementations. It s important to note that since ss,yy,zz only have physical properties inside the PML region, [SS] should be an identity tensor in normal computational domain i.e. σσ,yy,zz = outside the PML region, meanwhile gradually increased within the PML. The common method to achieve this is polynomial grading as: σ, y, z ( n) m n = σma d (3.31)

20 17 Where nn is the depth within the PML region, dd is the total depth of PML, σσ mmmmmm is a maimum conductivity when nn = dd. For many FDTD simulations, mm = 3~4 has found to be optimal, and an optimal choice for maimum conductivity can be epressed as [15] [17]: σ opt m π,, ( yz) (3.32) 3.6 Update Equation with PML The general time-harmonic form of Faraday s and Ampere s law in Eq. ( ) can be written as: E = j H ( ω) ω [ µ ] ( ω) H = j E + E ( ω) ω[ ε ] ( ω) [ σ] ( ω) (3.33) (3.34) To avoid computational errors caused by the high order of magnitude difference between electric and magnetic fields, Taflove [15] introduced a normalization of the magnetic field: (3.35) H = µ H ε Our new Mawell s equations with normalized magnetic fields and PML tensor incorporated in becomes: [ µ r ] [ ] ( ) E ( ) ω = jω S H ω c (3.36) [ ε ] r [ ] ( ) [ ] ( ) H ( ω) = jω S E ω+ σηe ω (3.37) c Then the two-dimensional TM mode vector epansions of Eq. ( ) are: ( ω) ( ω) Ez E y µ = jω y z c s sh 1 y ( ω) (3.38)

21 18 ( ω) E ( ω) E µ z = jω z c yy ss H 1 y y ( ω) (3.39) ( ω) H ( ω) H y ε ω ( ω) σ η ( ω) y c zz = j sse y z + zz Ez (3.4) After replacing ss,yy,zz with 1 + σσ,yy,zz /jjjjεε and denotes CCCC, CCCC yy as the curl of EE in, yy direction and CCCC zz as the curl of HH in zz direction, these equations become: y ( ω) = ω ( ω) 1 µ σ σ CE j H c jωε jωε µ σ σ CEy j H c jωε jωε yy y ( ω) = ω ( ω) ε σ σ CH E j E zz y ( ω) = ησ ( ω) + ω ( ω) z zz z z c jωε jωε 1 y (3.41) (3.42) (3.43) They can then be reformed into: σ c σ c jωh ω + H ω = CE ω CE ω (3.44) y ( ) ( ) ( ) ( ) ε µ jωε µ σ c c σ jωh ω + H ω = CE ω CE ω (3.45) y ( ) ( ) ( ) ( ) y y y y ε µ yy jωε µ r σ + σ σσ σ jωε E ω + E ω + E ω + E ω = c CH ω (3.46) y y zz ( ) ( ) ( ) ( ) ( ) zz z z 2 z z z ε jωε ε These equations can be transformed from frequency domain into time domain functions with a reverse Fourier transform as: ( ) t H t σ y c cσ + H ( t) = CE( t) CE( t) dt t ε µ εµ (3.47)

22 19 ε ( ) t Hy t σ c c σ y + H y( t) = CEy( t) CEy( t) dt t ε µ yy εµ yy (3.48) ( ) ( ) t zz z y zz y + z( ) + z + 2 z = z t ( ) t E t σ + σ σ t σσ E t E ( t) E ( t) dt c CH ( t) ε ε ε (3.49) Note that εε zzzz (tt) and σσ zzzz (tt) in Eq. (3.49) are functions of time because our simulations involve oscillating devices that have time-varying properties, the details will be illustrated in Section 4.3. Recall that in the Yee grid, there is a half temporal difference between the electric and magnetic fields. For instance, EE components at time tt have their corresponding HH components sampled at time tt tt/2 and tt + tt/2, thus HH components at time tt should be estimated by averaging the values of tt tt/2 and tt + tt/2. Let s specify that in Eq. ( ), the curl of EE eists at time tt and the curl of HH eists at tt + tt/2, that were previously calculated through spatial approimations in Eq. ( ). Then apply temporal central-difference approimations for derivatives in Eq. ( ) and estimate integrals with summations yields: t t t t H t+ H t H t H t σ + + c cσ t CE t CE T t 2 2 y = t ε 2 µ εµ T = ( ) ( ) (3.5) t t t t H t+ H t H t H t σ + + c cσ t CE t CE T y y y y t = y t ε 2 µ yy εµ yy T = y ( ) ( ) (3.51) ( t+ t) + ( t) E ( t+ t) E ( t) σ + σ E ( t+ t) + E ( t) ( t+ t) + ( t) ε ε y σ σ z z z z t ε 2 2ε zz zz zz zz y E z ( t)

23 2 ( ) ( ) σσ t y t Ez t+ t + Ez t t = + ε 4 T = 2 Ez( T ) cch z t (3.52) By eamining the above equations, it is evident that future field components HH (tt + tt/2), HH yy (tt + tt/2) and EE zz (tt + tt) can be calculated with current field components and known variables, thus the update equation for them can be derived as: H t 1 σ y t c cσ t H t CE t CE t t t 2ε 2 µ εµ T = t+ = 2 1 σ y + t 2ε ( ) ( ) (3.53) H y t 1 σ t c cσ t H y t CEy t CEy t t t 2ε 2 µ yy εµ yy T = t+ = 2 1 σ + t 2ε y ( ) ( ) (3.54) ( + ) + ( ) ( + ) + ( ) εzz t t εzz t σ + σ y σzz t t σzz t σσ y t 2 Ez ( t) + 2 t 2ε 2ε 4ε Ez ( t+ t) = ε ( t+ t) + ε ( t) σ + σ σ ( t+ t) + σ ( t) σσ t zz zz y zz zz y 2 2 t 2ε 2ε 4ε σσ t t c CH t E T y ( ) ( ) z 2 ε T = z (3.55) To further simplify our FDTD algorithm formulation, the non-time varying terms are collected as coefficients: 1 σ y mh = + t 2ε 1 σ y mh1 = / mh t 2ε

24 21 mh c µ 2 = / mh mh cσ t εµ 3 = / mh 1 σ mhy = + t 2ε mhy 1 σ mhy1 = / mhy t 2ε c µ 2 = / yy mhy cσ y t mhy3 = / mhy εµ yy σ + σ σσ mez1 = + 2ε 4ε y y t 2 σσ mez2 = ε y t 2 Since they are primarily comprised of constants and parameters that do not change once set, it would be much more computationally efficient to precompute them beforehand, store them as a lookup table that can be easily accessible during the simulation. The new update equation becomes: t t t H t + = mh1h t mh2ce( t) mh3 CE( T ) (3.56) 2 2 T = t t t H y t + = mhy1h y t mhy2cey( t) mhy3 CEy( T ) (3.57) 2 2 T = ( t+ t) + ( t) ( t+ t) + ( t) εzz εzz σzz σzz mez1 Ez ( t) + 2 t 2ε Ez ( t+ t) = εzz ( t+ t) + εzz ( t) σzz ( t+ t) + σzz ( t) + + mez1 2 t 2ε t t cch z t + mez2 E T z T = 2 ( ) (3.58) With update equations for all components in Faraday s law and Ampere s law, an FDTD simulation process can be illustrated in Figure.3 which forms the foundation of our simulations.

25 22 Figure 3. Flowchart of FDTD simulation in a 2-D TM mode 3.7 Other Conditions for FDTD Simulations For our spatial and temporal estimations, it s important that a sufficiently small δδ for central-difference approimations was chosen, it has been verified that at least ten cells per wavelength are necessary to ensure an acceptable accuracy for spatial approimations, although a higher resolution may be needed for more comple devices. Once a spatial size has been chosen, the temporal difference needs to be determined by the Courant-Friedrichs- Lewy stability condition [2]: t c + + y z (3.59) A common choice for tt to ensure stability and accuracy on any grid design is given as: min(, y, z) t = (3.6) 2c This epression also has the benefit that fields will travel eactly one grid cell in two timesteps in free space.

26 Source Ecitations There are different types of source signals suited for different purposes, two are used in this project, Gaussian pulse and sinusoidal continuous ecitation. A Gaussian function can be epressed as: ( ) ( t t ) 2 2 tω g t e = (3.61) Where tt is the temporal delay and tt ωω is the half-width of the Gaussian function. Typically, tt is chosen to be greater than 4tt ωω to avoid the simulation start inside the pulse. As for the choice of tt ωω, the Gaussian pulse has the characteristic of simulating a board range of frequencies tops at 1/ ππtt ωω ff mmmmmm 2/ππtt ωω, thus for a maimum frequency, a half-width could be chosen as tt ωω =.5/ff mmmmmm in this range. A sinusoidal source as a function of time can be written as sin(2ππππππ), however, it cannot be turned on instantaneously, rather be turned on with a smoothing function at the beginning, in our case a half Gaussian pulse. This can be achieved by creating a regular Gaussian pulse same as Eq. (3.61) and replace the value for tt > tt with ones: ( ) s t ( ) ( π ) ( π ) g t sin 2 ft < t t = for 1 sin 2 ft t > t (3.62) Where tt should not only be the peak of the Gaussian but also one of the peaks the sine function. This can be accomplished by choosing a small integer number nn and set tt = (4nn + 1)/4ff. 3.9 Total-field/Scattered-field Formulation For problems that employ a plane wave ecitation, directly modelling the source would be a substantial computational burden since it must eist far beyond the problem domain for

27 24 the wave fronts being parallel when they reach the devices. Thus, a method for injecting already paralleled waves directly into the problem domain is required. The most opportune one was proposed in [21], now commonly referred to as the total-field/scattered-field (TF/SF) formulation, assumes that the actual physical total electric and magnetic fields EE tttttttttt and HH tttttttttt can be split into: E = E + E total inc scat H = H + H total inc scat (3.63) (3.64) Where the incident-fields EE iiiiii and HH iiiiii are fields propagating through the problem domain without any interaction with simulated devices. The scattered-fields EE ssssssss and HH ssssssss are fields initially unknown and only affected by wave scattered by simulated devices. In reality, the boundary between the total-field and scattered field serves as an absorbing boundary condition that only absorbs ecitation waves. Figure.4 illustrates the total-field and scattered-field in the Yee grid. Consider the positions that have jj = jj where the scattered-field components EE zz and HH yy are located. Recall that the curl of EE on direction eists at the same spatial location as HH which reside in the total-field, to compute the CCCC ii,jj with our spatial approimations in Eq. (3.18), we need ii,jj the corresponding EE zz at (ii, jj ) and (ii, jj + 1). However, EE zz,tttttttttt does not eist because (ii, jj ) is located inside the scattered-field, therefore, CCCC ii,jj cannot be directly calculated. Yet, since E = E + E (3.65) i, j i, j i, j, total, scat, inc CCCC ii,jj can still be calculated with:

28 CE i, j ( ) E E E E + E E E E = = = y y y y i, j+ 1 i, j i, j + 1 i, j i, j i, j+ 1 i, j i, j z, total z, total z, total z, scat z, inc z, total z, scat z, inc (3.66) 25 ii,jj Note that the first two terms are equivalent to Eq. (3.18), which means CCCC can be computed normally without considering the TF/SF condition then subtract the incident terms at jj = jj : CE i, j Eq. ( 3.18 ) Ez, inc = CE (3.67) y i, j i, j ii,jj Similarly, CCCC zz located in the scattered-field can be calculated with HH located at (ii, jj + 1/2) inside the total-field and Eq. (3.2) as: CH i, j z + ( ) i j i j Hy, scat H H H H y, scat = y i, j 1/2 i, j 1/2 i, j 1/2, total, scat, inc + 1/2, 1/2, + ( ) i, j Eq = + CH z H i, j + 1/2. 3.2, inc y (3.68) ii,jj Now consider the situation on the opposite side where jj = jj 1. To compute CCCC 1 1 and ii,jj CCCC 1 zz requires EEzz in the scattered-field (ii, jj 1 ) and HH in the total-field (ii, jj 1 1/2). ii,jj Similar to jj = jj, CCCC 1 1 ii,jj and CCCC 1 zz can be computed normally then add or subtract the incident term at the boundary as: CE, 1 1 i, j1 1 i j i, j1 Eq. ( 3.18 ) Ez, inc = CE + (3.69) y CH ( ) H = CH y i, j1 i, j1 z z i, j1 + 1/2 Eq. 3.2, inc (3.7) For ii = ii, EE zz ii,jj in the scattered-field is required to calculate CCCC yy ii,jj, and CCCC zz ii,jj can be computed with HH yy ii +1/2,jj in the total-field as:

29 26 Figure 4. Field components in the 2-D TM mode grid at the boundaries of the total-field and scattered-field regions for plane wave eaction. Note the total-field is the region bounded by dashed lines. And regions outside are scattered-field. CE i, j y E E E = + i+ 1, j i, j i, j z, total z, scat z, inc (3.71) CH i, j z H H H H H = y i+ 1/2, j i 1/2, j i, j+ 1/2 i, j 1/ i+ 1/2, j y, total y, scat, scat, scat y, inc (3.72) ii They also can be calculated by computing our CCCC,jj ii yy and CCCC,jj zz normally with Eq. ( ) without considering TF/SF condition then correct with incident terms at ii = ii. CE i, j Eq. ( 3.19 ) Ez, inc = CE + (3.73) i, j i, j y y CH Eq. ( 3.2) H = CH i, j i, j z z i + 1/2, j y, inc (3.74) At ii = ii 1, EE zz ii 1,jj in the scattered-field is required to compute CCCC yy ii 1 1,jj, CCCC zz ii 1,jj can be computed with HH yy ii 1 1/2,jj in the total-field:

30 27 CE i1, j y i + 1, j 1 i, j Eq. ( 3.19 ) z, inc = CE (3.75) y E CH Eq. ( 3.2) H = CH + i1, j i1, j z z i1 1/2, j y, inc (3.76) Together, these corrections properly separate the total-field and scattered-field for our twodimensional TM mode grid, note that different placements of the TF/SF boundary will result in slightly different formulations. After implementing these corrections, a numerical plane wave can be generated in the total-field region propagating through the total-field region and being perfectly absorbed once hit the TF/SF boundary. By positioning the incident plane right net to a boundary, fields propagate toward that boundary will be absorbed immediately, thus appear as a plane wave propagating in the opposite direction. In addition, the boundaries between total-field and scattered-field are transparent to all outgoing scattered waves, permitting them to pass without reflection or refraction. 3.1 Correction Terms for TF/SF The crucial part of the TF/SF formulation is the correction terms. As previously defined, the incident-field is the source signal propagating through problem domain without interacting with simulated devices. One way of doing this is to simulate an identical secondary grid alongside our main grid with nothing ecept PML. In this project, the plane wave source is defined as propagating along yy ais. For the interface where jj = jj, since a plane wave only changes in the direction of its propagation, and this interface is norm to yy ais, all the incident terms on this interface will have the same value, the same also goes to the interface where jj = jj 1. As for interface where ii = ii and ii = ii 1, because the incident wave does not interact with anything, their value should be changing identically, only one of them need to be computed. Since the incident wave propagates in yy direction, there are

31 28 no component contributes to our HH field, therefore, our secondary grid can be reduced to a one-dimensional grid that can reuse our eisting formulations.

32 29 4. FDTD with an Oscillating Object In this section, a brief introduction of Raman scattering effect is outlined. And then a detailed description of the methodology used to incorporate an oscillating cylinder in our FDTD algorithm will be given. 4.1 Raman Effect When light incident on a small system, most of it will be scattered without any change in frequency. This is called Rayleigh Scattering after physicist Lord Rayleigh. A fraction of the scattering is inelastic and referred to as Raman scattering. Such scattering with a change in frequency was discovered by C.V. Raman in 1928 [22]. A detailed description of the effect could be found in [23], but to summarize: When Light is incident onto a system, it induces a dipole moment pp, which can be described as: p = a E (4.1) Where aa is the polarizability of the system, EE is the electric field of the incident signal traveling at frequency ff with an amplitude of EE, for a sinusoidal signal it can be given by: E = E sin 2 ft ( π ) (4.2) A vibrating system with frequency ff vv will change the polarizability of itself, resulting in the polarizability oscillating at ff vv : ( π ) a= a + βsin 2 ft (4.3) v Where aa is the polarizability in equilibrium configuration, and ββ is the variation of polarizability associated with system s vibration. Combine Eq. ( ), the induced dipole becomes:

33 3 ( π ) β ( π ) ( π ) p = aesin 2 ft + Esin 2 ft sin 2 fvt (4.4) Appling the trigonometric relations sin(aa) sin(bb) = [cos(aa BB) cccccc (AA + BB)]/2 it becomes: 1 p = aesin( 2π ft) + βe cos 2π ( f f ) t cos 2π ( f + f ) t 2 { v v } (4.5) This indicates the oscillating dipole will not only radiate light at frequency ff but also radiate light weakly at ff + ff vv and ff ff vv which give rise to Raman scattering. ff ff vv is called the Stokes frequency shift, ff + ff vv is referred to as the Anti-Stokes frequency shift. To mimic this effect, an oscillating cylinder will be modeled for our FDTD simulation in the net section. 4.2 Implementing Oscillating Device In FDTD simulations, devices within the problem domain are defined by specifying permittivity, permeability and conductivity values of each grid that represent the devices. For most materials, permeability is set to free-space since the material is not magnetizable, thus permeability will not be discussed further outside the FDTD formulations. To realize changes of a device s profile, only need to change the permittivity and conductivity values of the grid. An oscillating cylindrical rod in a two-dimensional grid can be represented with a circle that oscillates into ellipses. Consider an ellipse equation in a two-dimensional coordinate system, the area within it could be epressed as: ( ) ( y y ) a or b ( ) ( y y ) b (4.6) a Where (, yy ) is the center of the ellipse, aa and bb are its major and minor aes for which we define aa bb. And once aa = bb is satisfied, this function will also represent a circle. By oscillating the value of aa and bb periodically, the device can be modeled with periodic

34 31 oscillation. Given an oscillation frequency ff vv, the period would be pp = ff 1 vv, then this period needs to be divided into four fractions. For the first quarter, that is, any time tt that satisfies the condition of tt mmmmmm pp pp/4, our device should gradually transform from a circle with radius rr = aa+bb to an ellipse as a function of time: 2 ( ) O t ( ) ( ) ( ) ( ) y y = + r + dt r dt (4.7) Where tt = tt mmmmmm pp is a periodic time-step, dd is a small variation in length that equals 4 to 4dd mmmmmm /pp, and dd mmmmmm is a maimum deformation, thus have the relations: r + dt = a r dt = b for p p 3p tmod p=,,, p (4.8) For the second quarter that has pp/4 tt mmmmmm pp pp/2, our device should transform from the ellipse back to its equilibrium form with a function: ( ) O t ( ) ( ) ( ) ( ) y y = + a dt b + dt (4.9) In the third quarter that has pp/2 tt mmmmmm pp 3pp/4, the device deforms into ellipse same as in the first section but in the opposite direction: ( ) O t ( ) ( ) ( ) ( ) y y = + r dt r + dt (4.1) And at last, the device returns to its equilibrium in the fourth quarter period: ( ) O t ( ) ( ) ( ) ( ) y y = + b + dt a dt (4.11)

35 32 These formulations could be implemented numerically by creating two matries XX, YY representing the coordinate system, each row of XX is a full copy of all coordinates, i.e. XX(ii, : ) = ii for ii =1, 2, 3 and each column of YY contains the copy of all y coordinates YY(:, jj) = jj. Then by applying these formulations, the logic operator <= which correspond to less than equal will return a logical matri with coordinates (ii, jj) outside the devices marked as false and those within the device mark as true. Since true and false are stored as one and zero in computer memory, the coordination information contained in this matri can be straightforwardly turned into permittivity and conductivity matries as: ( t) = ( ) O( t) + ( t) ( ) O( t) ε ε ε ε r r1 r2 r2 σ = σ σ + σ (4.12) r1 r2 r2 Where O is the logical matri previously mentioned, and every note in the grid that represents our device has a relative permittivity of εε rr1 and conductivity of σσ rr1, yet other positions have relative permittivity εε rr2 and conductivity σσ rr2 which were set to one and zero to represent frees pace in this project. 4.3 Update Equation with Time-varying Property By oscillating our device, these properties become time-varying instead of constant compared to Yee s formulation and most common FDTD algorithms. An independently time-varying permittivity and conductivity can be written as: [ ε] = εε ( ) [ σ] σ ( t) r t Replace the terms in Eq. (3.9) result in: = (4.13) E H = εε r ( t) + σ( t) E t (4.14)

36 33 Recall that the curl of magnetic field eists at the same temporal position as magnetic field that has a half temporal difference with electric field, hence for the curl of HH at given time tt its corresponding EE field and electric properties should eist at tt + /2, therefore: t E t t + 2 t t CH ( t) εε r t = + + σ t + E t + 2 t 2 2 (4.15) Same as the electric field, the permittivity and conductivity at time tt + /2 also can be estimated by averaging the values at tt and tt + tt. Thus, after applying temporal approimation for our electric field on zz direction, the formulation becomes: ( ) CHz t ( t+ t) + ( t) E ( t+ t) E ( t) ( t+ t) + ( t) E ( t+ t) + E ( t) ε ε σ σ = ε + 2 t 2 2 zz zz z z z z (4.16) Which will result in our Eq. (3.52) formulation once perfectly matched layer is incorporated. However, there will be a drawback in achieving oscillation by changing the grid properties. Abrupt alteration in the state of the medium will affect the characteristics of fields eist within this medium. Specifically, this happens on the boundary of our device that changes its relative permittivity when it epands or contracts. 4.4 Transitional-layer To address the boundary problem, an additional transition-layer was implemented on the boundary of our device, aiming at drastically smoothening the variation of permittivity, so the change of permittivity becomes slow enough to be considered as time-invariant. Recall the formulation of our oscillation in Eq. ( ), since these formulations eist on a grid system, all the variables should be integers ecept dd which equals to 4dd mmmmmm /pp, the ellipse

37 34 would only change once per dd 1 time-steps when dd tt reaches an integer. By anticipating how the ellipse would change net time, the dd 1 time-steps can be utilized to integrate linearly increasing or decreasing functions to make permittivity alterations much smoother during the simulation. Consider our formulation of the first periodical section Eq. (4.7), after dd 1 time-steps the function becomes: ( ) ( ) ( ) ( ) y y O( t+ d ) = + r + dt + 1 r dt 1 (4.17) Where the oscillation is epanding in direction and contracting in yy direction, subtracting OO(tt + dd 1 ) by O( t ) yields a matri with coordinate information of changes between two time-steps. As illustrated in Figure.5, points in area E would be zero indicate invariant, points in A and B would have a value of one indicate epansion, and points inside C and D have a value of negative one indicates contraction. Note that Figure.5 is not an accurate representation, A, B, C, D would be a thin layer of grids that have the thickness of one grid. The permittivity change in respect of time can be characterized as a function: Figure 5. Representation of ellipse oscillating during first quarter of the period. A, B are epanding, C, D are contracting.

38 35 1 t mod d ε ε ε d ( t) = ( ) 1 r2 r1 (4.18) Our permittivity function with transition-layer build-in then becomes: 1 ( t) = ( ) O( t) + + ( t) O( t+ d ) O( t) ε ε ε ε ε r r1 r2 r2 (4.19) With a relatively slow oscillation compared to the incident frequency and a reasonable oscillation amplitude, the relative permittivity variation can be effectively stretch to at least -3 orders of magnitude per time-step without noticeable stability issues.

39 36 5. Results With the system described in previous sections, several simulations were designed to validate if the system is working as epected and giving us some insight on how the electromagnetic fields react to an oscillating device. The geometry of the simulation was set up as in Figure.6. An oscillating device is located in the middle of the grid and surrounded by the TF/SF boundaries, the device is inside the total-field and an observation point is located in the scattered field perpendicular to the y ais. A sinusoidal plane wave source was incident net to the boundary in the total-field, so the incident wave will only travel in y + direction. The oscillatory motion will start after the continuous wave filled the problem domain. Fields passing through the observation point will be recorded during the simulation, so the spectrum of the field can be obtained using Fourier transform after the system reaches a steady state. By changing the parameters of the system such as the incident frequency, diameter of the device, etc., how the fields interact with different configurations will be studied. Figure 6. Schematic of the simulation setup, the computational domain is represented by the most outside solid line, area between the solid line and the blue dash line is 1 cells deep PML. Total-field/Scattered-field boundaries are surrounding the oscillating device in the middle. And the incident position net to the Total-field/Scattered field boundary.